The complex numbers corresponding to the endpoints of one diagonal of a square drawn on a complex plane are 1 + 2i and -2 – i.

what are the complex numbers corresponding to the endpoints of the square's other diagonal?

It would help to plot these on your complex plane.
It's pretty much just like a normal coordinate plane, but instead of the y-axis you have your imaginary axis.
1+2i and -2-i are the equivalents of (1, 2) and (-2, -1).

The slope between these points is 1. (3 rise over 3 run which simplifies to 1)
Since our slope has that 45° elevation, it's going to be paralel to our axes.
We can just find the endpoints of our diagoanal (the other corners of our square) by taking the real part of one coordinate (so it's put vertically from one point) and the imaginary part of the other. (so it's put horizontally from the other)
If you don't understand what I mean, try graphing.

Our other points would correspond to the complex numbers are 1-i and -2-2i.

Use pythagorean’s theorem and find the missing side lengths of the two smaller triangles and add them together
13^2 - 12^2 = 5
15^2 - 12^2 = 9
5+9 = 14
14 is the length of each side of the square
14^2 + 14^2 = 19.8
19.8 is the diagonal length of the inside if the square